An efficient hybrid method for modeling lipid membranes with molecular resolution G.J.A. (Agur) Sevink, M. Charlaganov & J.G.E.M Fraaije Japan, 2010 Soft Matter Chemistry Group, Leiden University, The Netherlands Thanks to : C.D. Chau, A.V. Zvelindovsky, organizers!
Outline Motivation Japan, 2010 Hybrid CG modeling (ongoing, conceptual) Enhanced sampling (quick) Conclusions/outlook
Motivation Using/adapting dynamic mesoscopic methodology for block copolymers to simulate life-mimicking (biomematic) structures and structure formation Japan, 2010 • Veterinarian • Biologist • Physicist • Mathematician ?
Motivation reduction Japan, 2010 “Cell membrane dynamics essentially lipidic” (100+ simulation papers) VW Project 2009-2012 ‘Multiscale hybrid modeling of (bio)membranes’ (Schmid, Zvelindovsky, Böker, AS) Aim: Realistic computational modeling of liposome formation, dynamics and (assisted) fusion
Motivation: intriguing experiments in Leiden Japan, 2010 Vesicle fusion induced by coiled-coil motif (short peptide fragments) Hana Robson Marsden et al, A reduced SNARE model for membrane fusion, Angew. Chem. 2330–2333, 2009.
General issues: length and time scales Nm and mm: model for complete vesicle and/or vesicle fusion requires considerable coarse graining Efficient, realistic, dynamic Japan, 2010 micellar growth collapse growth (coalescence) fast slow fast Ostwald ripening closure & vesicle fusion and fission J. Leng, S. Egelhaaf, M. Cates fast extremely slow (2002) Europhys. Lett.
The DNA of simulation Methods Based on SDSC Blue Horizon (SP3) 512-1024 processors 1.728 Tflops peak performance CPU time = 1 week / processor 10 0 Atomistic Mesoscale methods Thermodynamics (ms) 10 -3 Simulation Methods Japan, 2010 Average, collective ( µ s) 10 -6 Semi-empirical (ns) 10 -9 methods Monte Carlo Statistical physics molecular dynamics (ps) 10 -12 Ab initio Specific, detail methods tight-binding MNDO, INDO/S (fs) 10 -15 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4
Vesicle formation and fusion (2005) 100 nm METHOD: DDFT= mean-field SCFT+diffusion 20% A 2 B 2 in a selective bad solvent Japan, 2010 Movie 200 nm AS., Zvelindovsky A.V. Macromolecules 38 7502-7513 (2005).
Issues Beyond block copolymers: How to realistically represent lipids? Japan, 2010 Increasing complexity? ‘Floppy’ Gaussian chains: onion vesicles Mean-field: concentrated systems
Hybrid particle-field model Japan, 2010 Aim: flexibility, efficient and realistic liposome simulation (ongoing work)
DDFT: pattern formation dynamics in concentrated BCP Enthalpic: mean-field interactions (FH) 2 F [ ρ I ] = F ideal [ ρ I , U I ] + F cohesive [ ρ I ] + 1 ∑ ∫ 2 κ H ρ I V I Pressure term, incompressible Entropic: Gaussian chains in self- Japan, 2010 consistent field U noise d ρ I ( r ) = M ∇ ⋅ ρ I ( r ) ∇ δ F [ ρ , U ] ( r ) + .......... + η I ( r ) dt δρ I local kinetic model processing conditions (quasi)equilibrium behavior, AB, ABC, branced Phase transition under external fields (confinement, shear, E, etc)
Synergetic validation: flat polymeric ‘membrane’ Structural transition due to thickness reduction : top view Experiment Japan, 2010 Δ t sim ~ sec Calculation annihilation splitting nucleation High-speed SFM measurements of membrane dynamics: ~ sec ptf
Different representations of constituents Variable composition solvent chemical fragment spring Japan, 2010 DDFT : Underlying harmonic spring, calculations and interactions field-based Particles (DPD) : Harmonic spring, angle and torsion potentials, soft core repulsive pair potentials f repulsive 0 + Δ a ij a ij = a ij a ij liquid incompressibility r distance c
Hybrid model F hybrid [ ρ I , k ] = F DDFT [ ρ I ] + U particles [ k ] + F coupling [ ρ I , r r r k ] K ( r − r k ) ρ I ( d Japan, 2010 ∑ ∫ c Ik r ) r V I , k particles conserv − ∑ ∫ random ( t ) ∂ r k = D k [ f k c Ik K ( r − r k ) ∇ ρ I ( r ) dr ] ∂ t + r k V I d ρ I ( = M ∇ ⋅ ρ I ( r ) ) ∇ [ µ DDFT ( r ) + ∑ r c Ik K ( r − r k ) ] + η I dt k fields Diffusion, timescales are more or less comparative (coupled update)
Physical interpretation ρ Ι ( r ) Positive c r Coupling force: away from high density field values Japan, 2010 Coupling chemical potential: field diffuses away from regions with many particles Advantage is possibility to mix different representations on CG level for same or different constituents: sparse (particles) + abundant (field) Mapping: besides FH parameter ( χ )/interaction strengths (a) we need c Ik κ compressibility ( ) and coupling ( ).
Mapping particles and fields: binary system Determine ‘free’ parameters by requiring thermodynamic consistency for single bead solvent in both representations. κ : match either pressure or c Ik Japan, 2010 κ excess chemical potential c Ik : use field partitioning to determine FH χ and Groot & Warren to convert to soft-core potential strength → c Ik = c Ik ( a ) Note : both particles and fields adapt dynamically
Hybrid vs DPD lipid membrane simulation (16 3 ) Use these values and realistic DPD lipid parameters solvent field solvent particles Japan, 2010 solvent field solvent particles Hybrid calculation where the solvent is DPD, Shillcock and Lipowky 2002 replaced by a field, with the same S&L (realistic) parameters for the lipid
Hybrid membrane simulation Japan, 2010 Averaging over many initial condition and time frames
Additional benefits: implicit solvent Preliminary: analytical equilibrium solution for solvent (field) can be converted into an additional potential in particle description ( sol , ( ρ sol , lipid ) lipid ) lipid r r r r → → k k k k mapping analytic Japan, 2010 V → A CGMD, implicit solvent I.R. Cooke, K. Kremer, M. Deserno, Phys. Rev. E, 011506 (2005).
Vesicle formation pathway following quench solvent field Japan, 2010 300000 200000 Diffusion is patient (DPD – O(20000)) 32 3 Experiments: slow process! Solution? S-QN: accelerating collective modes
Japan, 2010 Enhanced sampling: Accelerating collective modes in a CG particle description
Stochastic Quasi-Newton method Optimization in numerical mathemetics (objective function) Δ x k = x k + 1 − x k = − α k ∇Φ Steepest descent Δ x k = x k + 1 − x k = − α k H − 1 ∇Φ Newton method B k → H − 1 Japan, 2010 B k Quasi-Newton method Diffusion in statistical mechanics (potential function) Δ x k = − M ∇Φ ( x k ) Δ t + 2 Mk B T Δ t Δ W k M(x) Curvature-dependent mobility √ M(x) M ( x ) = ( ∇ 2 Φ ( x )) − 1 Fluctuation-dissipation + spurious drift
Stochastic Quasi-Newton method Illustration: 1-D Harmonic oscillator Φ ( x ) ~ k 2 x 2 for M = 1 dx = − kxdt + 2 k B TdW ( t ) 2 k B T M = k − 1 = ( ∇ 2 Φ ) − 1 for dx = − xdt + k dW ( t ) Japan, 2010 drift term noise term ~ k -1 slow modes fast modes k<1 k>1 Sparse sampling Dense sampling Δ t max Stability analysis: independent of k
Stochastic Quasi-Newton method approximate of H ( x k ) − 1 M ( x ) = M k ( x k ) = M k ( x k ,..., x 0 ) New factorized update method (equivalent to DFP) for M k+1 : M k + 1 − M k F • Hereditary: minimal Japan, 2010 • If M 0 positive definite, M k+1 positive definite ( √ M exists!) • M k+1 is approximate of inverse Hessian (secant condition) T • Efficiency: update J k+1 M k + 1 = J k + 1 J k + 1 Rouse chain Additional costs per timestep but Δ t SQN >> Δ t LD M k → H −
Stochastic Quasi-Newton method Analysis for quadratic potential (Rouse chain): all modes evolve equally fast (real-space Fourier acceleration) Minimal model of a protein Φ = 1 2 Φ bond + 1 Japan, 2010 2 Φ bending + Φ dihedral + Φ LJ Bead=amino acid (either neutral, hydrophobic or hydrophilic) Conclusions (S-QN): Enhanced sampling of energy landscape (many inherent states) Hierarchical optimization (bond length, angles, torsions, non bonded) Generic S-QN method: accelerated but no ’realistic’ dynamics
Conclusions and outlook Conclusions: New hybrid model for particle/field mixtures Reuse DPD parameters for CG lipids Possibility of implicit solvent (analytic) Additional sparse constituents can be added as CG particle chains New S-QN method to speed up formation kinetics Japan, 2010 To do: Validate membrane material parameters in hybrid model Concise derivation of implicit solvent Implementation and parameterization of SNARE-like CG proteins Outlook: Large scale simulations Vesicle fusion …
… Thank you for your attention Japan, 2010 Questions? (a.sevink@chem.leidenuniv.nl)
Japan, 2010
T > T collapse Minimal model of a protein (3D): sampling efficiency Standard LD (SLD) One basin Several basins Our FSU method ‘native state’ Principles of SQN Principles of SQN 29 29
T << T fold Minimal model of a protein (3D): mode analysis LB 8 B(NL) 2 NBLB 3 LB Native state: left, turn and right sub-domains χ χ SLD FSU native Φ = 1 2 Φ bond + 1 2 Φ bending + Φ dihedral + Φ LJ Equilibration order: bonds, angles, torsions, LJ (even for reduced spring constants) -> ‘soft’ RATTLE/SHAKE/LINCS Principles of SQN Principles of SQN 30 30
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